Nonequilibrium stretching dynamics of dilute and entangled linear polymers in extensional flow

We propose an extension of the FENE-CR model for dilute polymer solutions [M.D. Chilcott, J.M. Rallison, Creeping flow of dilute polymer solutions past cylinders and spheres, J. Non-Newtonian Fluid Mech. 29 (1988) 382–432] and the Rouse-CCR tube model for linear entangled polymers [A.E. Likhtman, R.S. Graham, Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie–Poly equation, J. Non-Newtonian Fluid Mech. 114 (2003) 1–12], to describe the nonequilibrium stretching dynamics of polymer chains in strong extensional flows. The resulting models, designed to capture the progressive changes in the average internal structure (kinked state) of the polymer chain, include an ‘effective’ maximum contour length that depends on local flow dynamics. The rheological behavior of the modified models is compared with various results already published in the literature for entangled polystyrene solutions, and for the Kramers chain model (dilute polymer solutions). It is shown that the FENE-CR model with an ‘effective’ maximum contour length is able to describe correctly the hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation also observed and computed by Doyle et al. [P.S. Doyle, E.S.G. Shaqfeh, G.H. McKinley, S.H. Spiegelberg, Relaxation of dilute polymer solutions following extensional flow, J. Non-Newtonian Fluid Mech. 76 (1998) 79–110] and Li and Larson [L. Li, R.G. Larson, Excluded volume effects on the birefringence and stress of dilute polymer solutions in extensional flow, Rheol. Acta 39 (2000) 419–427] using Brownian dynamics simulations of bead–spring model. The Rolie–Poly model with an ‘effective’ maximum contour length exhibits a less pronounced hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation.     

On the dynamics of grafted branched polymers—a Monte Carlo simulation study

A simplified model of grafted branched polymers was designed and investigated. The model consisted of star-branched chains constructed on a simple cubic lattice. The star polymers were built of three arms of equal lengths. The chains were attached to an impenetrable flat surface with one arm’s end. The arm attached to the surface (a stem) was built of segments different from those in two remaining arms (branches). During the Monte Carlo simulation of the system, the conformation of each chain was modified according to the metropolis sampling algorithm with local changes of chain conformations. The simulations were performed for different chain lengths and the temperature of the system (solvent conditions). The structure of a polymer film formed on the grafting surfaces depended strongly on the temperature and the low temperature films consisted of two separate layers with the insoluble layer located near the grafting surface. The short-time relaxation of the branches and stems of chains was also investigated. The analysis of the dynamics of the model system shows the influence of the structure of the system on relaxation times of various parameters. Paper presented at the AERC 2005 held on April 21–23, 2005 in Grenoble, France.     

Birefringence and stress growth in uniaxial extension of polymer solutions

Simultaneous measurements of extensional stresses and birefringence are rare, especially for polymer solutions. This paper reports such measurements using the filament stretch rheometer and a phase modulated birefringence system. Both the extensional viscosity and the birefringence increase monotonically with strain and reach a plateau. Estimates of this saturation value for birefringence, using Peterlin’s formula for birefringence of a fully extended polymer chain are in agreement with the experimental results. However, estimates of the saturation value of the extensional viscosity using Batchelor’s formula for suspensions of elongated fibres are much higher than observed. Reasons for the inability of the flow field to fully unravel the polymer chain are examined using published Brownian dynamics simulations. It is tentatively concluded that the polymer chain forms a folded structure. Such folded chains can exhibit saturation in birefringence even though the stress is less than that expected for a fully extended molecule.Simultaneous measurements of stress and birefringence during relaxation indicate that the birefringence decays much more slowly than the stress. The stress-birefringence data show a pronounced hysteresis as predicted by bead-rod models. The failure of the stress optic coefficient in strong flows is noted.Experiments were also performed wherein the strain was increased linearly with time, then held constant for a short period before being increased again. The response of the stress and birefringence in such experiments is dramatically different and can be traced to the different configurations obtained during stretching and relaxation. The results cast doubt on the appropriateness of pre-averaging the non-linear terms in constitutive equations.     

Elongational flow and rheology of monodisperse polymers in solution

We describe the utilization of idealized stagnation point extensional flows, produced by opposed jets, for birefringence visualization of induced molecular strain and flow resistance measurements. We identify rheological changes associated with the coil---stretch transition which occurs beyond a critical strain-rate in elongational flow-fields. In dilute solutions of monodisperse atactic polystyrene, increases in extensional viscosity are observed as isolated molecules become stretched. The largest increases in extensional viscosity, however, are found only beyond a critical concentration and strain rate, and are associated with the stretching of transient networks of interacting molecules. These results parallel similar effects seen in porous media flow and capillary entrance experiments. We determine the molecular weight dependence of the critical concentration which scales as M^−0.55 in agreement with pairwise interaction of coils, but is much lower than conventional values of the critical polymer concentration, c^*. We believed that polydispersity may play an important role in the development of such transient networks, and in controlling the degradation behaviour during flow.     

Computer simulation of the rheology of concentrated star polymer suspensions

We use particle-based computer simulations to study the rheology of suspensions of high-functionality star polymers with long entangled arms. Such particles have properties which are intermediate between those of soft colloidal particles and entangled polymer chains. In the simulations, each star polymer is coarse-grained to a single particle. In order to faithfully reproduce dynamical properties, it is very important to not only include time-averaged interactions (potentials of mean force) but to also account for transient interactions induced by entanglements between the arms of different star polymers. Using a model which has all these features, it is found that, for sufficiently high shear rates, the start-up shear stress displays an overshoot. With increasing concentration, the core interactions increasingly dominate the initial stress response, leading to a maximum in the stress overshoot at relatively low strain values (0.1 to 0.5). Transient forces start to dominate after this initial stage. In a simulated experiment in which the shear rate is suddenly stepped-down from a high to a lower value, the stress shows a clear undershoot, with the minimum stress again at a relatively low strain value (based on the new shear rate). Finally, it is shown that a stress plateau develops in the flow curve. This plateau is absent when the transient forces between the polymer stars are not taken into account.     

Mesoscopic dynamics of polymer chains in high strain rate extensional flows

We take a step towards accessing the physics of viscoelastic liquid breakup in high speed, high strain rate flows by performing Brownian dynamics computations of dilute uniaxial, equibiaxial, and ellipsoidal polymeric extensional flows. Our computational implementation of the bead-spring model, when tailored to the DNA molecule, consistently with recent works of Larson and co-workers, is shown: (a) to predict a coil-stretch transition at Deborah number De≈0.5, and (b) to reproduce the experimental longest relaxation time. Furthermore, after adapting the model parameters to represent the polyethylene oxide (PEO) chain (for M=10⁶ Da), we find it possible to reproduce our own experimental data of the longest relaxation time, the transient extensional viscosity of dilute solutions at small Deborah numbers, and a coil-stretch transition at Deborah number De≈0.5. Extended to large Deborah numbers, the model predicts that polymer stretching is controlled by: (a) the randomness of the initial conditions that, in combination with rapid kinematically imposed compression, lead to the formation of initially frozen chain-folds, and (b) the speed with which thermo-kinematic processes relax these folds. The slowest fold relaxation occurs during uniaxial extension. As expected, the introduction of stretching along a second direction enhances the efficiency of fold relaxation mechanisms. Even for Deborah numbers (based on the chain longest relaxation time) of the order of one thousand, there is a large variation in the time a polymer needs in order to extend fully, and the effects of Brownian motion cannot be ignored. The computed Trouton ratios and polymer contributions to the total stress as functions of Hencky strain provide information about the relative importance of elastic effects during polymeric liquid stretching. At high strain rates, the steady state elastic stresses increase linearly with the Deborah number, resembling at that stage an anisotropic Newtonian fluid (constant extensional viscosity).     

Primitive chain network simulations for branched polymers

We present simulations of branched polymer dynamics based on a sliplink network model, which also accounts for topological change around branch points, i.e., for branch-point diffusion. It is well-known that, with the exception of stars, branched polymers may show a peculiar rheological behavior due to the exceptionally slow relaxation of the backbone chains bridging branch points. Though Brownian simulations based on sliplinks are powerful tools to study the motion of polymers and to predict rheological properties, none of the existing methods can simulate the relaxation of the bridge chains. The reason for that is lack of a rule for network topology rearrangement around branch points, so that entanglements between bridge chains cannot be renewed. Therefore, we introduce in this paper one possible branch-point mobility rule into our primitive chain network model. For star polymers, diffusion coefficients were calculated and compared with experiments. For both star and H-shaped polymers, diffusion was simulated both with and without the new rule, and the effect on linear viscoelasticity was also determined in one case.     

Extensional viscosity in uniaxial extension and contraction flow—Comparison of experimental methods and application of the molecular stress function model

Extensional viscosity of a low-density polyethylene was measured at three temperatures in uniaxial extension by Sentmanat Extension Rheometer, and in contraction flow using the Cogswell analysis. The molecular stress function model was applied to describe the experimental results. The achieved maximum values from uniaxial transient tests were in accordance with the ones obtained by Cogswell method at similar strain level, and the molecular stress function model was able to describe the experimental transient uniaxial extensional data. The steady-state extensional viscosity was not reached in the experiments.     

Birefringence measurements in extensional flow of polyisobutylene around an expanding bubble

Birefringence in liquid polymers offers the possibility of obtaining information about stress in complex flows. In this work, this is done for extensional flows of polyisobutylene in a “breathing bubble” rheometer. In this type of rheometer, a bubble consisting of an incompressible, low-viscosity fluid (usually water) is injected into the sample with a nozzle. Expanding or collapsing the bubble by adding or removing water induces biaxial or uniaxial extension in the surrounding sample. The pressure difference between the bubble and the surroundings can be measured and compared to the predictions of constitutive equations. This measurement only gives one integral value for a complex flow history. In this paper, the birefringence around the bubble is measured in order to learn more about the flow. This is done by comparing pressure and birefringence results to those of standard constitutive equations for a polyisobutylene sample. A good agreement between the pressure and optical measurements and the theory is found with a single value of the stress-optical constant. Received: 25 June 1997 Accepted: 12 November 1997     

Extensional behavior influence on viscoelastic turbulent channel flow

In this work we use in the simulation of a viscoelastic turbulent channel flow a modification of the finitely extensible of non-linear elastic dumbbells with the Peterlin approximation (FENE-P) constitutive model for dilute polymer solutions, applicable to high extensional deformations. The new feature introduced by this modification is that the free energy of the polymer (since it is assumed to be entirely entropically driven) remains always bounded (FENE-PB). The characteristics of the model under steady shear flow, pure elongational flow and transient extensional behavior are presented. It is found that the FENE-PB model is more shear thinning than FENE-P. Most importantly, it also shows a higher extensional viscosity than the FENE-P model. Although the steady-state Trouton ratio asymptotically reaches at high extensional rates the same limit as the FENE-P model, the transition from the Newtonian value is sharper and faster. We use the FENE-PB model in direct numerical simulations (DNS) of viscoelastic turbulent channel flow using spectral approximations. The results for various statistics of the flow and the polymer conformation, when compared against those obtained with the original FENE-P model and the same rheological parameters, show an enhanced polymer-induced drag reduction effect and enhanced deformation of the polymer molecules. This indicates that it is not only the asymptotic but also details from the extensional rheological behavior that matter in quantitatively specifying turbulent viscoelastic flow behavior.     

Flow of wormlike micelle solutions through a periodic array of cylinders

Solutions of self-assembled wormlike micelles are used with ever increasing frequency in a multitude of consumer products ranging from cosmetic to industrial applications. Owing to the wide range of applications, flows of interest are often complex in nature; exhibiting both extensional and shear regions that can make modeling and prediction both challenging and valuable. Adding to the complexity, the micellar dynamics are continually changing, resulting in a number of interesting phenomena, such as shear banding and extensional flow instabilities. In this paper, we present the results of our investigation into the flow fields generated by a controllable and idealized porous media: a periodic array of cylinders. Our test channel geometry consists of six equally spaced cylinders, arranged perpendicular to the flow. By systematically varying the Deborah number, the flow kinematics, stability and pressure drop were measured. A combination of particle image velocimetry in conjunction with flush mount pressure transducers were used to characterize the flow, while flow induced birefringence measurements were used to determine micelle deformation and alignment. The pressure drop was found to decrease initially due to the shear thinning of the test fluid, and then exhibit a dramatic upturn as other elastic effects begin to dominate. We present evidence of the onset of an elastic instability in one of the test fluids above a critical Deborah number manifest in fluctuating transient pressure drop measurements and asymmetric streamlines. We argue that this disparity in the two test fluids can be attributed to the measurable differences in their extensional rheology.     

Flow-induced nonequilibrium thermodynamics of lamellar semicrystalline polymers

In this work we present a first attempt to quantify the effect of flow deformation on the microstructure of semicrystalline polymers. This necessitates bridging the macroscopic flow length scale with the microscopic (segment) length scale of the semicrystalline structure. To achieve this connection we developed a hierarchical approach where a thermodynamically consistent macroscopic constitutive equation is interfaced with a microscopic lattice-based Monte Carlo (MC) simulation of the polymer chain conformation. We first illustrate this approach in a two-dimensional (2D) “toy” application where the 2D equivalent of a macroscopic constitutive equation based on reptation theory is applied to describe the chain deformation and extended free energy in the amorphous bulk phase. The values for the derivative of the free energy with respect to the mean segment orientation tensor, calculated for a planar extensional flow, are then used as an extended nonequilibrium thermodynamic forcing term. This is added in a traditional Metropolis Monte Carlo scheme, developed for a 2D lattice representation of a lamellar semicrystalline polymer, to drive the flow-induced microstructure. Significant flow-induced changes are calculated, steadily increasing as the Weissenberg number increases.We subsequently extend these ideas further in a much more realistic three-dimensional (3D) application where the information for the thermodynamics of the bulk amorphous phase under a uniaxial extensional flow is extracted from a macroscopic network model, such as that of Phan-Thien and Tanner (PTT), connecting the free energy to the second moment of the end-to-end distance of a multisegment chain. Through a series of 3D nonequilibrium Monte Carlo simulations of both the amorphous and the semicrystalline microscopic morphologies, it is shown that the interaction of the flow-induced deformation with the semicrystalline microstructure is nonlinear: the amorphous interlamellar structure changes significantly from its corresponding homogeneous bulk amorphous state, even far away from the crystalline interface. Our approach allows for a quantitative estimation of this effect on both thermodynamic quantities, like the extended microscopic free energy, as well as various statistics of the chain conformations.     

Brownian configuration fields and variance reduced CONNFFESSIT

Stochastic simulation techniques, such as Brownian dynamics, provide us an extremely powerful tool for solving the usually nonlinear equations describing polymer dynamics in solutions and melts [1]. However, the most challenging problems (e.g. the investigation of the universal behaviour of long polymer chains, or the flow calculation based on stochastic simulation techniques) involve a very large number of degrees of freedom and hence require an enomous amount of computer time. In order to solve such problems on currently available computers it is therefore necessary to develop strategies to drastically suppress the level of the fluctuations in the simulations. The purpose of this note is to show that the recently proposed concept of Brownian configuration fields [2] in viscoelastic flow calculations can be regarded as an extremely powerful extension of variance reduction techniques based on parallel process simulation.     

Flowing colloidal suspensions in non-Newtonian suspending fluids: Decoupling the composite birefringence

The rheology of dilute, colloidal suspensions in polymeric suspending fluids can be studied with simultaneous dichroism and birefringence measurements. The dichroism provides a direct measure of the particle dynamics, but the birefringence is a composite property with independent contributions from the suspended particles and the polymer molecules. For suspensions where the contribution from the particles is significant, the composite birefringence must be decoupled in order to analyze the dynamics of the polymeric suspending fluid. A method to perform the decoupling is derived and then demonstrated through transient shear flow experiments with dilute suspensions ofFeOOH particles in semi-dilute, xanthan gum suspending fluids. The birefringence of the xanthan gum suspending fluid is calculated from experimental measurements of the composite birefringence and the dichroism of the suspension. To gather information on particle/polymer interactions, the calculated birefringence is compared to the birefringence of xanthan gum solutions containing no suspended particles and the dirchoism is compared to that of a suspension in a Newtonian fluid.     

Purely elastic instabilities in three-dimensional cross-slot geometries

Creeping and low Reynolds number flows of an upper-convected Maxwell (UCM) fluid are investigated numerically in a three-dimensional orthogonal cross-slot geometry. We analyze two different flow configurations corresponding to uniaxial extension and biaxial extension, and assess the effects of extensional flow type, Deborah and Reynolds numbers on flow dynamics near the interior stagnation point. Using these two flow arrangements the amount of stretch and compression near the stagnation point can be varied, providing further insights on the viscoelastic flow instability mechanisms in extensionally dominated flows with an interior stagnation point. The uniaxial extensional flow arrangement leads to the onset of a steady flow asymmetry, followed by a second purely elastic flow instability that generates an unsteady flow at higher flow rates. On the other hand, for the biaxial extension flow configuration a symmetric flow is observed up to the critical Deborah number when the time-dependent purely elastic instability sets in, without going through the steady symmetric to steady asymmetric transition.     

Macroscopic theory of orientation transitions in the extensional flow of side-chain nematic polymers

A macroscopic continuum mechanical model for incompressible side-chain nematic polymers, under isothermal conditions is given. The model is a synthesis of a transient network model and the standard nematorheological model. Simplifications in the model yield constitutive equations that are identical to well known Theological models for polymer melts and for low molar mass nematics. A detailed analysis of four possible composite orientation modes of polymer backbone and mesogenic side groups in uniaxial extensional flow is given. It is shown that the thermal sensitivity of the viscoelastic parameters leads to thermally-induced orientation transitions. The extension rate sensitivity of the competition between elastic and flow orienting effects leads to flow-induced orientation transition. The role of smectic A fluctuations in thermally-induced transitions during uniaxial extensional nematic flow is elucidated. The model is able to predict and explain the experimentally observed orientation modes and thermally-induced orientation transitions of a side-chain nematic polymer subjected to uniaxial extensional flow.     

A solution birefringence study of polymer elongation in spatially periodic stretching flows

The flow birefringence of dilute polymer solutions in periodically converging/diverging channels has been employed to study the dynamics of flexible chain molecules under transient stretching stresses. The onset of periodic birefringence for chains of high molecular weight is only observed after the chains have experienced several cycles of stress, at a point deep into the channel. This slow onset indicates that the solutions possess a memory on time scales much greater than that normally associated with a relaxed flexible coil. This behavior is recorded for both aqueous and nonaqueous solutions, at concentrations both above and below the entanglement concentration. Centerline birefringence, however, which is associated with purely elongational flow, is only observed for aqueous polymer solutions. An explanation for these birefringence results is suggested, based on a configuration-dependent dumbbell model for polymer chains in dilute solution.     

On the Kinetics of Polymer Crystallization in Opposite-Nozzle Flow

The present work was encouraged by the successes obtained previously in this laboratory with short-term shearing experiments on slightly undercooled melts of i-PP: post-shearing lamellar growth on (inconspicuous) thread-like precursors. For the present purpose (evaluation of the influence of extensional flow) the pioneering work by Mackley and Keller is taken as the point of departure. Our own machine of the same type has been adapted for creep experiments (adjustment to steady flow in fractions of the time needed in the original machine). The range of extension rates, where a transition takes place from a mere multiplication of the number of nuclei to the induction of highly oriented structures, appears to be quite narrow in undercooled i-PP melts. In the range of high extension rates (≅50 s⁻¹ ) the critical time for the formation of an oriented structure could not be measured because of its shortness (less than 0.2 s). It turns out that the flow pattern in the opposite-nozzle machine is far from ideal. A proposal had to be made for a redesign. In spite of the preliminary nature of some of our results, several interesting insights should not be “bottled up”. First of all, there is the usefulness of creep flow (because of its fast transition into steady state, after an almost instantaneous compliance). Secondly, there is the quite unexpected ineffectiveness of lower stretching rates for the formation of oriented structures. Thirdly, there is the overwhelming influence of a change of the geometry: the provisional introduction of trumpet-shaped (nearly hyperbolic) entrance regions to the nozzles caused a remarkable broadening of the birefringent zone, which was previously observed as a very thin “string” connecting the nozzles. Finally, the almost certain usefulness of the revised machine for other (sometimes purely rheological) purposes, e.g., for steady-state flow birefringence measurements in extensional flow should be mentioned. Received: 22 June 1999 Accepted: 28 September 2000     

哑铃式聚合物分子模型流变性质的Brown动力学模拟

采用非平衡态Ｂｒｏｗｎ动力学方法（ＮＥＢＤ）模拟ＦＥＮＥ哑铃分子模型在定常拉伸流动和突然开始拉伸流动中的运动，计算流变性质，考虑位形相关Ｓｔｏｋｅｓ阻力系数等模型参数的影响，并从非线性随机动力学角度，分析相图，给出拉伸粘度曲线和哑铃伸展曲线不会出现Ｓ型的解释